
Like [1]}, a dense matrix is constructed by \(A = U\Sigma V^T\in ^{m\times n}\) ,
where the entries of \(U\in ^{m\times r}\)  and \(V\in ^{n\times r}\)  are generated from a standard normal distribution,
and then their columns are orthonormalized. The matrix \(\Sigma \)  is an \(r\) -by-\(r\)  diagonal matrix whose first \(r\) -2 diagonal entries are uniformly distributed numbers in \( (\sigma _2, \sigma _1) \) , and the last two diagonal entries are \(\sigma _2\)  and \(\sigma _1\) . The synthetic matrix is abbreviated as \( A = {\sf Smatrix} (m, n,r,\sigma _1,\sigma _2)\) .
